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Schismatic family: Difference between revisions

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<span style="display: block; text-align: right;">Other languages: [[:de:Schismatische_Temperaturen|Deutsch]]</span>
<span style="display: block; text-align: right;">Other languages: [[:de:Schismatische_Temperaturen|Deutsch]]</span>
The 5-limit parent comma for the '''schismatic family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields &lt;&lt;1 -8 -15|| for the [[wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768.


= Five limit =
= 5-limit =
The 5-limit parent comma for the schismatic family is the [[schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus comma]] ([[81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo|-15 8 1}}, and flipping that yields &lt;&lt;1 -8 -15|| for the [[Wedgies_and_Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
 
The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called '''Helmholtz''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.


[[POTE generator]]: ~3/2 = 701.736
[[POTE generator]]: ~3/2 = 701.736
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Map: [&lt;1 0 15|, &lt;0 1 -8|]
Map: [&lt;1 0 15|, &lt;0 1 -8|]


EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[65edo|65]], [[77edo|77]], [[94edo|94]], [[118edo|118]], [[171edo|171]], [[183edo|183]], [[224edo|224]], [[289edo|289]], 407, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc
{{EDOs|legend=1| 12, 29, 41, 53, 65, 77, 94, 118, 171, 183, 224, 289, 407, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc }}


Badness: 0.00426
Badness: 0.00426
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Eigenmonzos: 2, 9/7
Eigenmonzos: 2, 9/7


[[POTE_tuning|POTE generator]]: ~3/2 = 702.085
[[POTE generator]]: ~3/2 = 702.085


Mapping generator: ~3
Mapping generator: ~3
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Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||
Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||


EDOs: 12, 29, 41, 53, [[94edo|94]], 241c, 335cd, 576cd
{{EDOs|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576cd }}


Badness: 0.0216
Badness: 0.0216
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Eigenmonzos: 2, 9/7
Eigenmonzos: 2, 9/7


[[POTE_tuning|POTE generator]]: ~3/2 = 702.157
[[POTE generator]]: ~3/2 = 702.157


Mapping generator: ~3
Mapping generator: ~3
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Map: [&lt;1 0 15 25 -33|, &lt;0 1 -8 -14 23|]
Map: [&lt;1 0 15 25 -33|, &lt;0 1 -8 -14 23|]


[[EDO|Edos]]: [[41edo|41]], [[53edo|53]], [[94edo|94]], [[135edo|135]], 229c, 323c, 417ce
{{EDOs|legend=1| 41, 53, 94, 135, 229c, 323c, 417ce }}


Badness: 0.0274
Badness: 0.0274
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Commas: 225/224, 275/273, 325/324, 385/384
Commas: 225/224, 275/273, 325/324, 385/384


[[POTE_tuning|POTE generator]]: ~3/2 = 702.113
[[POTE generator]]: ~3/2 = 702.113


Mapping generator: ~3
Mapping generator: ~3
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Map: [&lt;1 0 15 25 -33 -28|, &lt;0 1 -8 -14 23 20|]
Map: [&lt;1 0 15 25 -33 -28|, &lt;0 1 -8 -14 23 20|]


EDOs: 12e, 41, 53, 94, 135, 429cdef, 523cdef
{{EDOs|legend=1| 12e, 41, 53, 94, 135, 429cdef, 523cdef }}


Badness: 0.0207
Badness: 0.0207
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[[Category:Rank 2]]
[[Category:Rank 2]]


{{todo|review}}
{{todo|review|improve readability}}
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